Iterate over all letters, First letter is added c1 = a1 times, each other letter is added ci = min(ai, ci - 1). Don't forget that if some letter is not added at all, then all next letters are not added too.

623A - Graph and String Note that all vertices "b" are connected with all other vertices in the graph. Find all such vertices and mark them as "b". Now we need to find any unlabeled vertex V, mark it with "a" character. Unlabeled vertices connected with V should be also labeled as "a". All other vertices we can label as "c"

Finally we need to check graph validity. Check that all vertices "a" are only connected with each other and "b" vertices. After that we need to perform a similar check for "c" vertices.

At least one of ends (a1 or an) is changed by at most 1. It means that if gcd > 1 then it divides on of prime divisors of either a1 - 1, a1, a1 + 1, an - 1, an or an + 1. We will iterate over these primes.

Suppose prime p is fixed. For each number we know that it's either divisible by p or we can pay b to fix it or it should be in the subarray to change for a

We can use dynamic programming dp[number of numbers considered][subarray to change not started/started/finished] = minimal cost

Complexity is O(Nd) = O(Nlog(max(ai)), where d is the number of primes to check.

First of all consider cases where all points are projected to the same axis. (In that case answer is difference between maximum and minimum of this coordinate).

Now consider leftmost and rightmost points among projected to x axis. Let xL and xR are their x-coordinates. Notice that points with x-coordinate xL ≤ x ≤ xR may also be projected to x-axis and that will not increase the diameter. So, if we sort all points by x-coordinate, we may suppose that points projected to x-axis form a continuous subarray.

We will use a binary search. Now we will need to check if it's possible to project point in a such way that diameter is <= M.

Let's fix the most distant by x-coordinate point from 0 that is projected to x-axis. It may be to the left or to the right of 0. This cases are symmetrical and we will consider only the former one. Let xL < 0 be its coordinate. Notice that one may project all points such that 0 ≤ x - xL ≤ M and |x| ≤ |xL| to the x axis (and it'll not affect the diameter) and we have to project other points to y-axis. Among all other points we should find the maximum and minimum by y coordinate. Answer is "yes (diam ≤ M)" if ymax - ymin < = M and distance from (xL, 0) to both (0, ymax) and (0, ymin) is not greater than M.

Let's precalculate maximums and minimums of y coordinates on each prefix and suffix of original (sorted) points array. Now iterate over left border of subarray of points projected to x-axis and find the right border using binary search or maintain it using two-pointers technique.

Main idea: first of all guess each friend once, then maximize probability to end game on current step. Let's simulate first 300000 steps, and calculate . , where ki — how many times we called i-th friend ().

Expectation with some precision equals . So it is enough to prove that:

1) Greedy strategy gives maximum values for all Pr(t).

2) On 300000 step precision error will be less than 10 - 6.

Proof:

1) Suppose, that for some t there exists set li (), not equal to set produced by greedy algorithm ki, gives the maximum value of Pr(t). Let's take some ka < la and kb > lb, it is easy to prove tgat if we change lb to lb + 1, la to la - 1, then new set of li gives bigger value of Pr(t), contradiction.

2) qi ≤ 0.99. Let's take set , it gives probability of end of the game not less than optimal. Then Pr(t) ≥ (1 - 0.99t / 100)100 ≥ 1 - 100·0.99t / 100. Precision error does not exceed . It could be estimated as sum of geometric progression. If N ≥ 300000 precision error doesn't exceed 10 - 7.

First observation is that if the sequence of prefix xors is strictly increasing, than on each step ai has at least one new bit comparing to the previous elements. So, since there are overall k bits, the length of the sequence can't be more than k. So, if n > k, the answer is 0.

Let's firstly solve the task with O(k3) complexity. We calculate dp[n][k] — the number of sequences of length n such that a1|a2|... |an has k bits. The transition is to add a number with l new bits, and choose those k bits which are already in the prefix xor arbitrarily. So, dp[n + 1][k + l] is increased by dp[n][k]·2k·Ck + ll. The last binomial coefficient complies with the choice these very l bits from k + l which will be present in a1|a2|... |an + 1.

Note now that the transition doesn't depend on n, so let's try to use the idea of the binary exponentiation. Suppose we want to merge two dynamics dp1[k], dp2[k], where k is the number of bits present in a1|a2|... |aleft and b1|... |bright correspondingly. Now we want to obtain dp[k] for arrays of size left + right. The formula is:

Here l corresponds to the bits present in the xor of the left part, and for each number of the right part we can choose these l bits arbitrarily. Rewrite the formula in the following way:

So, we can compute dp[k] for all k having multiplied two polynomials and . We can obtain the coefficients of the first polynomial from the coefficients of the second in . So, we can compute this dynamic programming for all lengths — powers of two, in , using the fast Fourier transform. In fact, it is more convenient to compute using the same equation. After that, we can use the same merge strategy to compute the answer for the given n, using dynamics for the powers of two. Overall complexity is .

We decided to ask the answer modulo 109 + 7 to not let the participants easily guess that these problem requires FFT :) So, in order to get accepted you had to implement one of the methods to deal with the large modulo in polynomial multiplication using FFT. Another approach was to apply Karatsuba algorithm, our realisation timed out on our tests, but TooDifficuIt somehow made it pass :)

The algorithm is greedy: first, take the minimal number with sum of digits a1 — call it b1. Then, on the i-th step take bi as the minimal number with sum of digits ai, which is more than bi - 1.

It can be easily proven that this algorithm gives an optimal answer. But how to solve the subproblem: given x and y, find the minimal number with sum of digits x, which is more than y?

We use a standard approach: iterate through the digits of y from right to left, trying to increase the current digit and somehow change the digits to the right in order to reach the sum of digits equal to x. Note that if we are considering the (k + 1)-th digit from the right and increase it, we can make the sum of k least significant digits to be any number between 0 and 9k. When we find such position, that increasing a digit in it and changing the least significant digits gives us a number with sum of digits x, we stop the process and obtain the answer. Note that if k least significant digits should have sum m (where 0 ≤ m ≤ 9k), we should obtain the answer greedily, going from the right to the left and putting to the position the largest digit we can.

Let us bound the maximal length of the answer, i.e. of bn. If some bi has at least 40 digits, than we take the minimal k such that 10k ≥ bi. Than between 10k and 10k + 1 there exist numbers with any sum of digits between 1 and 9k. If k ≥ 40, than 9k ≥ 300, which is the upper bound of all bi. So, in the constraints of the problem, bi + 1 will be less than 10k + 1. Than, similarly, bi + 2 < 10k + 2 and so on. So, the length of the answer increases by no more than one after reaching the length of 40. Consequently, the maximal length of the answer can't be more than 340.

The complexity of solution is O(n·maxLen). Since n ≤ 300, maxLen ≤ 340, the solution runs much faster the time limit.

First we note that if the sequences ai and bi are a valid solution, then so are the sequences ai - P and bi + P for any integer P. This means that we can consider a1 to be equal to 0 which allows us to recover the sequence bi by simply taking the first row of the matrix. Knowing bi we can also recover ai (for example by subtracting b1 from the first column of the matrix) At this stage we allow ai and bi to contain negative numbers, which can be later fixed by adding K a sufficient amount of times. Now we consider the “error” matrix e: .

If e consists entirely of 0s, then we’ve found our solution by taking a sufficiently large K. That is: K > maxi, j(wi, j).

Otherwise, we note that ei, j = 0(modK) which implies that K is a divisor of g = gcdi, j(ei, j). The greatest such number is g itself, so all that remains is to check if g is strictly greater than all the elements of the matrix w. If that is the case, then we’ve found our solution by setting K = g. Otherwise, there’s no solution.

We first calculate the prefix sums of vowel(si) which allows to calculate the sum of vowel(si) on any substring in O(1) time.

For all m from 1 to , we will calculate the sum of simple pretinesses of all substrings of that length, let’s call it SPm. For that purpose, let’s calculate the number of times the i-th character of the string s is included in this sum.

For m = 1 and m = |s|, every character is included exactly 1 time. For m = 2 and m = |s| - 1, the first and the last character are included 1 time and all other characters are included 2 times. For m = 3 and m = |s| - 2 the first and the last character are included 1 time, the second and the pre-last character are included 2 times and all others are included 3 times, and so on.

In general, the i-th character is included min(m, |s| - m + 1, i, |s| — i + 1) times. Note that when moving from substrings of length m to substrings of length m + 1, there are 2 ways in which the sum SP can change:

If m > |s| - m + 1, then SP is decreased by the number of vowel occurrences in the substring from |s| - m + 1 to m.

Otherwise, SP is increased by the number of vowel occurrences in the substring from m to |s| - m + 1.

This way we can easily recalculate SPm + 1 using SPm by adding (subtracting) the number of vowel occurrences on a substring (which is done in O(1) time). The complexity of this solution is O(N).

Consider a tree with n vertices rooted at vertex 1 and let b be the pseudocode’s (DFS) resulting sequence. Then b[lv..lv + sizev - 1], represents vertex v’s subtree, where lv is the index of v in b and sizev is the size of $v$’s subtree.

Let’s solve the problem using this fact and Dynamic Programming. Let e[l, r] be the number of trees consisting of vertices a[l], a[l + 1], …, a[r] such that running DFS starting from a[l] will result in a sequence with vertices in the same order as their order in a.

The base case is when l = r and e[l, r] = 1. Otherwise, where the sum is taken over all partitions of the segment [l + 1, r], that is, over all k;pos1, ..., posk + 1, such that l + 1 = pos1 < pos2 < ... < posk + 1 = r, 1 ≤ k ≤ r - l, a[pos1] < a[pos2] < ... < a[posk]. Each such partition represents a different way to distribute the vertices among a[l]’s children’s subtrees. A solution using this formula for calculating e[l, r] will have an exponential running time.

The final idea is to introduce d[l, r]: = e[l - 1, r], 2 ≤ l ≤ r ≤ n. It follows that: d[l, r] = ([statement] is equal to 1 if the statement is true, 0 otherwise) and e[l, r] = d[l + 1, r]. This way d[l, r] and e[l, r] can be calculated in linear time for any segment [l, r]. The answer to the problem is e[1, n]. Overall complexity is O(n3).

Support CLion 2016.3.2: return back to changing only settings for testrunner

0.10

Support CLion 2016.3

Switch configuration now changes global setting AND setting for testrunner

Rewrite task saving so that random exceptions are not thrown

0.9.2

Support AtCoder and CodeAcademy parsers

0.9

Now includes inlined when they point to inside project (not when they are in quotes as before)

JHelper is plugin for writing contests in C++. You may inline code from your own prewritten library so that you can submit only used code. Besides, it allows to test on tests you've added. it's planned that you'll be able to parse a problem/contest and have all samples tests automatically added.